Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x²≤9

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1. Rewrite the inequality \(x^2 \leq 9\) to isolate zero on one side: \(x^2 - 9 \leq 0\).
2. Factor the left-hand side using the difference of squares formula: \(x^2 - 9 = (x - 3)(x + 3)\), so the inequality becomes \((x - 3)(x + 3) \leq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\), and \(x + 3 = 0\) gives \(x = -3\). These points divide the number line into intervals.
4. Test values from each interval \((-\infty, -3)\), \((-3, 3)\), and \((3, \infty)\) in the inequality \((x - 3)(x + 3) \leq 0\) to determine where the product is less than or equal to zero.
5. Based on the test results, write the solution set in interval notation, including the points where the product equals zero since the inequality is 'less than or equal to'.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it means finding all x-values that satisfy the inequality, often by analyzing the related quadratic equation and determining where the parabola lies above or below a certain line.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation (e.g., x² = 9) to find critical points. These points divide the number line into intervals, which are tested to determine where the inequality holds true.

Interval Notation

Interval notation is a concise way to express solution sets of inequalities. It uses parentheses and brackets to indicate open or closed intervals, representing all numbers between endpoints that satisfy the inequality.